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Full text of "Analytical Mechanics"

UNIPLANAR MOTION OF A RIGID BODY          235
Therefore after the time k the hoop will roll along towards the right with a linear velocity v2 less than VQ, and with an angular velocity o> greater than OJQ.
Case III.  Suppose the initial rotation to be counter-clockwise.   In this case we obtain
V3 _ s^a,                         (9,0
where U is the time when sliding ceases.
There are three special cases to be considered :
(a)  When VQ > aa>0, vz is positive, and consequently the hoop goes on rolling towards the right.
(b)  When VQ = aco0, vs = 0, and consequently at t = tz the hoop comes to rest.
(c)  When VQ < oco0, v$ is negative.   Therefore at the instant t = t& the hoop begins to roll backwards.
PROBLEMS.
1.   Discuss the motion of the following bodies rolling down an inclined plane without slipping:
(a)  A hollow cylinder of mass m and inner and outer radii TI and r2j respectively.
(b)  A hoop of mass m and radius r.
(c)  A sphere of mass m and radius r.
(d)   A hollow sphere of mass m and inner and outer radii r^ and r2, respectively.
(e)   A spherical shell of negligible thickness of mass m and of radius r.
(f)   Compare the times of descent in (c) and (e).
2.   A sphere is projected, without initial rotation, up a perfectly rough inclined plane.    Discuss the motion.
3.   A wheel which is rotating about its own axis is placed on a perfectly rough inclined plane.    Discuss the motion up the plane. "